Galileo's Paradox
in [Math]
|
From: MoeBlee on 11 Dec 2006 13:51 Tony Orlow wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> You might want to look into Internal Set Theory, a partial > >> axiomatization of Nonstandard Analysis. > > > > Why do you say 'parital'? > > I said "partial", and said that because that's what I've read. I am not > sure what parts of nonstandard analysis are not included in the axioms > of IST. > > > > >> Both infinitesimal and infinite > >> values are "nonstandard", and no reference to "standard" values is > >> allowed in the definition of any set. > > > > Not ANY set. IST includes standard sets too. You do realize that IST is > > an EXTENSION of ZF, right? > > > > MoeBlee > > > > Sorry, "standard" is not allowed in the definition of any *internal* set. You didn't answer the question. IST is an extension of ZF. IST includes every theorem of ZF (plus more theorems). IST is not a theory that contradicts or even excludes ZF. IST is a theory of which ZF is INCLUDED. I'm just curious whether you know that, since you reject ZF but then I read you recommending that people look into IST. MoeBlee
From: MoeBlee on 11 Dec 2006 13:58 Tony Orlow wrote: > Look it up. Robinson's Nonstandard Analysis is THE first rigorous > treatment of infinitesimals, Robinson's work in non-standard analysis is in classical mathematical logic and set theory, with even the axiom of choice. > apparently, and Internal Set Theory is a > set-theoretic axiomatization of it, apparently. :) An axiomatization that includes all the axioms of ZF. MoeBlee
From: MoeBlee on 11 Dec 2006 14:01 Tony Orlow wrote: > > Fortunately, TO's opinion is of no weight. > > But it has a great value range. Tony makes an inside joke about his own Tonyology. No narcissist, he. Not much. MoeBlee
From: cbrown on 11 Dec 2006 14:09 Tony Orlow wrote: > cbrown(a)cbrownsystems.com wrote: > > (T1) infinite(x) <-> A yeR x>y > > Tony Orlow wrote: > > > >>>> infinite(x) <-> A yeR x>y > >>> Is that your only axiom? If so, then state your first theorem about them > >>> and give the proof. > >>> > >> That's the only one necessary for what defining a positive infinite n. A > >> whole array of theorems pop forth... > > > > Before going there, you might want to start by adding the axiom: > > > > (T2) exists B such that infinite(B) > > > > Otherwise, who cares if you can prove a whole bunch of theorems about > > something that doesn't exist? > > > > Cheers - Chas > > > > What do you mean by "exist"? That's what I get for letting sloppy notation confuse me :). I'll put it another way: When you assert "infinite(x) <-> Ay in R, x > y", what are we supposed to think you mean by "x > y"? For example, let T be an equilateral triangle with unit length sides. Is T > 1.72? Cheers - Chas
From: MoeBlee on 11 Dec 2006 14:41
Tony Orlow wrote: > > I have read Robinson. On what page of what book does he refer to omega - > > 1 in comparison to omega? I do not find any such reference. > > He uses the assumption that any infinite number can have a finite number > subtracted, and become smaller, like any number except 0, so there is no > smallest infinite, just like you do with the endless finites. > Non-Standard Analysis, Section 3.1.1: You are REPEATING the mistake you've made from the beginning regarding Robinson, because you never bothered to read it and understand it but instead just jumped around to find things you only THINK say what you think they should say. In CONTEXT, when Robinson is talking about those infinite numbers, he is NOT talking about ordinals such as omega. There are two DIFFERENT senses of 'infinite' in play, one is infinite cardinality and the other is that of a certain ordering of elements in certain sets, which is NOT a cardinality ordering and does NOT apply to omega the way you have answered. And I can predict you're going to say something along the lines of that you're not saying what I have said you said. But then your answer makes NO SENSE as a response about OMEGA. > "There is no smallest infinite number. For if a is infinite then a<>0, > hence a=b+1 (the corresponding fact being true in N). But b cannot be > finite, for then a would be finite. Hence, there exists an infinite > numbers [sic] which is smaller than a." > > Of course, he has no for omega. It's illegitimate schlock, like I said. What does, "he has no for omega" mean? What is illegitimate schlock? Robinson uses ZFC and classical mathematical logic all over the place. MoeBlee |